Laplace-based strategies for Bayesian optimal experimental design with nuisance uncertainty

Abstract

Finding the optimal design of experiments in the Bayesian setting typically requires estimation and optimization of the expected information gain functional. This functional consists of one outer and one inner integral, separated by the logarithm function applied to the inner integral. When the mathematical model of the experiment contains uncertainty about the parameters of interest and nuisance uncertainty, (i.e., uncertainty about parameters that affect the model but are not themselves of interest to the experimenter), two inner integrals must be estimated. Thus, the already considerable computational effort required to determine good approximations of the expected information gain is increased further. The Laplace approximation has been applied successfully in the context of experimental design in various ways, and we propose two novel estimators featuring the Laplace approximation to alleviate the computational burden of both inner integrals considerably. The first estimator applies Laplace's method followed by a Laplace approximation, introducing a bias. The second estimator uses two Laplace approximations as importance sampling measures for Monte Carlo approximations of the inner integrals. Both estimators use Monte Carlo approximation for the remaining outer integral estimation. We provide four numerical examples demonstrating the applicability and effectiveness of our proposed estimators.

Figures (10)

• Figure 1: Example 1: Optimal number of outer ($N^{\ast}$) and inner ($M_1^{\ast}$, $M_2^{\ast}$) samples vs. tolerance $TOL$ for the DLMC, DLMC2IS, and MC2LA estimators.
• Figure 2: Example 1: EIG vs. design parameter $\xi$. Analytical solution (dashed black), DLMC2IS estimator (solid blue), and MC2LA estimator (solid red) for the case with nuisance uncertainty and DLMCIS estimator (solid magenta) and MCLA estimator (solid tan) for the case without nuisance uncertainty.
• Figure 3: Example 1: Error vs. tolerance consistency plot for various tolerances $TOL$ with a predefined confidence parameter $C_\alpha=1.96$ ($\alpha=0.05$). Panel (A) DLMC2IS estimator. Panel (B) MC2LA estimator.
• Figure 4: Example 2: Optimal number of outer ($N^{\ast}$) and inner ($M_1^{\ast}$, $M_2^{\ast}$) samples vs. tolerance $TOL$ for the DLMC, DLMC2IS, and MC2LA estimators for geometrically spaced sampling times.
• Figure 5: Example 2: Geometrically spaced design God20 yielding expected information gain (EIG) of 6.12 vs. optimized design yielding EIG of 6.25. Clustering of measurement times appeared to have a positive impact on the EIG of the experiment. Later measurement times had little effect on the EIG and were mostly unaffected by the optimization.

Theorems & Definitions (5)

• Remark 1: Differences and additional expenses in accounting for nuisance uncertainty in the Laplace approximation of the posterior
• Remark 2: Uniqueness of the minimum
• Remark 3
• Remark 4
• Remark 5: Covariance terms in the statistical error approximation